Seminar

Prof. Heinz Hanssmann (University of Utrecht)

“Families of hyperbolic Hamiltonian tori”

Thursday 11 February 2016,  15.00

Dipartimento di Matematica (aula Dal Passo) Università di Roma “Tor Vergata”

Abstract:

In integrable Hamiltonian systems hyperbolic tori form families, parametrised by the actions conjugate to the toral angles. The union over such a family is a normally hyperbolic invariant manifold. Under Diophantine conditions a hyperbolic torus persists a small perturbation away from integrability. Locally around such a torus the normally hyperbolic invariant manifold is the centre manifold of that torus and persists as well. We are interested in `global’ persistence of the normally hyperbolic invariant manifold. An important aspect is how the dynamics behaves at the (topological) boundary. Where the manifold extends to infinity this boundary is empty – this case makes clear that we need the persistence theorem of normally hyperbolic invariant manifolds in the non-compact setting. If the normal hyperbolicity wanes as the boundary is approached we need to ensure that the perturbed dynamics does not come closer to the boundary. This provides the necessary uniform lower bound of normal hyperbolicity to still ascertain persistence under small perturbations. Making use of energy preservation and of Diophantine tori persisting by KAM theory this can be achieved for families of two-dimensional hyperbolic tori.

Thursday 12th November 2015, h 9:30 Sala Conferenze, Collegio Puteano, Centro De Giorgi, Pisa

Henk Bruin (University of Vienna)

“Sharp mixing rates via inducing with respect to general return times”

Abstract: For non-uniformly expanding maps inducing w.r.t. a general (i.e., not necessarily first) return time to Gibbs Markov maps, we provide sufficient conditions for obtaining sharp estimates for the correlation function. This applies to both the finite and the infinite measure setting. The results are illustrated by non-Markov intervals maps with an indifferent fixed point. This is joint work with Dalia Terhesiu.

Thursday  03/9/2015, h 16:30 Sala Conferenze (Puteano, Centro De Giorgi)

Mark Pollicott (University of Warwick)

“LINEAR RESPONSE AND PERIODIC ORBITS”

When: Wednesday July 22, 2015, at 11:30

Where: Seminario I, Dept. of Mathematics, Università di Bologna

Who: Francesco Cellarosi (Queen’s University, Canada)

What: Seminar “Recent progress towards Sarnak’s and Chowla’s Conjectures”

Abstract: I will present an overview of Sarnak’s conjecture on the disjointness of the Möbius function from any deterministic sequence and the related Chowla’s conjecture on the self-correlations of the Möbius function. Some progress towards weaker versions of these conjectures have been made recently, and I plan to illustrate them.

Monday  20/7/2015, h 11:30-12:30 Sala Conferenze (Puteano, Centro De Giorgi)

Jacopo De Simoi (University of Toronto)

“AN INTEGRABLE BILLIARD CLOSE TO AN ELLIPSE OF SMALL ECCENTRICITY IS AN ELLIPSE”

Abstract: In 1927 G. Birkhoff conjectured that if a billiard in a strictly convex smooth domain is integrable, the domain has to be an ellipse (or a circle). The conjecture is still wide open, and presents remarkable relations with open questions in inverse spectral theory and spectral rigidity. In the talk we show that a version of Birkhoff’s conjecture is true for small perturbations of ellipses of small eccentricity. This is joint work with A. Avila and V. Kaloshin

Friday, July 17, 2015, h. 16:00-17:00 Sala Conferenze (Puteano, Centro De Giorgi)

Sandro Vaienti (CPT Marseille)

“On recent results of extreme value theory applied to dynamical systems”

Abstract:  We give a review of a few recent results of extreme value theory applied to random dynamical systems and to coupled lattice maps.